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 A324525 Numbers divisible by prime(k)^k for each prime index k. 15
 1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These are a kind of self-describing numbers (cf. A001462, A304679). A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization. Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA Closed under multiplication. Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020 EXAMPLE The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.     1: {}     2: {1}     4: {1,1}     8: {1,1,1}     9: {2,2}    16: {1,1,1,1}    18: {1,2,2}    27: {2,2,2}    32: {1,1,1,1,1}    36: {1,1,2,2}    54: {1,2,2,2}    64: {1,1,1,1,1,1}    72: {1,1,1,2,2}    81: {2,2,2,2}   108: {1,1,2,2,2}   125: {3,3,3}   128: {1,1,1,1,1,1,1} MAPLE q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]): select(q, [\$1..3000])[];  # Alois P. Heinz, Mar 08 2019 MATHEMATICA Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k>=PrimePi[p]]&] seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *) CROSSREFS Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679. Cf. A109298, A324524, A324571, A324572, A324587, A324588. Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360. Sequence in context: A335851 A028982 A320137 * A175338 A071601 A114400 Adjacent sequences:  A324522 A324523 A324524 * A324526 A324527 A324528 KEYWORD nonn AUTHOR Gus Wiseman, Mar 08 2019 STATUS approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)